A Posteriori Error Estimators for the Frank-Oseen Model of Liquid Crystals

TL;DR

This paper develops and analyzes a posteriori error estimators for the Frank-Oseen liquid crystal model, enabling efficient adaptive mesh refinement to improve solution accuracy and computational efficiency.

Contribution

It introduces reliable a posteriori error estimators for the nonlinear Frank-Oseen model with unit-length constraints, applicable to both Lagrange multiplier and penalty methods.

Findings

Adaptive mesh refinement reduces computational cost.

Error estimators effectively guide mesh adaptation.

Solutions on adapted meshes are comparable to uniform refinement.

Abstract

This paper derives a posteriori error estimators for the nonlinear first-order optimality conditions associated with the Frank-Oseen elastic free-energy model of nematic and cholesteric liquid crystals, where the required unit-length constraint is imposed via either a Lagrange multiplier or penalty method. Furthermore, theory establishing the reliability of the proposed error estimator for the penalty method is presented, yielding a concrete upper bound on the approximation error of discrete solutions. The error estimators herein are composed of readily computable quantities on each element of a finite-element mesh, allowing the formulation of an efficient adaptive mesh refinement strategy. Four elastic equilibrium problems are considered to examine the performance of the error estimators and corresponding adaptive mesh refinements against that of a simple uniform refinement scheme. The adapted grids successfully provide significant reductions in computational work while producing solutions that are highly competitive with those of uniform mesh in terms of constraint conformance and computed free energies.